In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, the no-deleting theorem of
quantum information theory
Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information re ...
is a
no-go theorem
In theoretical physics, a no-go theorem is a theorem that states that a particular situation is not physically possible. Specifically, the term describes results in quantum mechanics like Bell's theorem and the Kochen–Specker theorem that cons ...
which states that, in general, given two copies of some arbitrary quantum state, it is impossible to delete one of the copies. It is a time-reversed
dual to the
no-cloning theorem In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theore ...
, which states that arbitrary states cannot be copied. This theorem seems remarkable, because, in many senses, quantum states are fragile; the theorem asserts that, in a particular case, they are also robust. Physicist
Arun K. Pati along with
Samuel L. Braunstein
Samuel Leon Braunstein (born 1961) is a professor at the University of York, UK. He is a member of a research group in non-standard computation, and has a particular interest in quantum information, quantum computation and black hole thermody ...
proved this theorem.
The no-deleting theorem, together with the no-cloning theorem, underpin the interpretation of quantum mechanics in terms of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, and, in particular, as a
dagger symmetric monoidal category
In the mathematical field of category theory, a dagger symmetric monoidal category is a monoidal category \langle\mathbf,\otimes, I\rangle that also possesses a dagger structure. That is, this category comes equipped not only with a tensor product ...
. This formulation, known as
categorical quantum mechanics
Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably monoidal category theory. The primitive objects of study are physical processes, and the diff ...
, in turn allows a connection to be made from quantum mechanics to
linear logic
Linear logic is a substructural logic proposed by Jean-Yves Girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. Although the logic has also be ...
as the logic of
quantum information theory
Quantum information is the information of the quantum state, state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information re ...
(in exact analogy to classical logic being founded on
Cartesian closed categories
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in ma ...
).
Overview of quantum deletion
Suppose that there are two copies of an unknown quantum state. A pertinent question in this context is to ask if it is possible, given two identical copies, to delete one of them using quantum mechanical operations? It turns out that one cannot. The no-deleting theorem is a consequence of linearity of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. Like the no-cloning theorem this has important implications in
quantum computing
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
,
quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both th ...
theory and
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
in general.
The process of quantum deleting takes two copies of an arbitrary, unknown
quantum state at the input port and outputs a blank state along with the original. Mathematically,
this can be described by:
:
where
is the deleting operation which is not necessarily unitary (but a linear operator),
is the unknown quantum
state,
is the blank state,
is the initial state of
the deleting machine and
is the final state of the machine.
It may be noted that classical bits can be copied and deleted, as can
qubits
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
in orthogonal states. For example, if we have two identical
qubits
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
and
then we can transform to
and
. In this case we have deleted the second copy. However, it follows from linearity of quantum theory that there is no
that can perform the deleting operation for any arbitrary state
.
Formal statement of the no-deleting theorem
Let
be an unknown
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
in some
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
(and let other states have their usual meaning). Then,
there is no linear isometric transformation such that
, with the final state of the ancilla being independent of
.
Proof
The theorem holds for quantum states in a Hilbert space of any dimension. For simplicity,
consider the deleting transformation for two identical qubits. If two qubits are in orthogonal states, then deletion requires that
:
,
:
.
Let
be the state of an unknown qubit. If we have two copies of an unknown qubit, then by linearity of the deleting transformation we have
:
:
In the above expression, the following transformation has been used:
:
However, if we are able to delete a copy, then, at the output port of the deleting machine, the combined state should be
:
.
In general, these states are not identical and hence we can say that the machine fails to delete a copy. If we require that the final output states are same, then we will see that there is only one option:
:
and
:
Since final state
of the ancilla is normalized for all values of
it must be true that
and
are orthogonal. This means that the quantum information is simply in the final state of the ancilla. One can always obtain the unknown state from the final state of the ancilla using local operation on the ancilla Hilbert space. Thus, linearity of quantum theory does not allow an unknown quantum state to be deleted perfectly.
Consequence
* If it were possible to delete an unknown quantum state, then, using two pairs of
EPR states, we could send signals faster than light. Thus, violation of the no-deleting theorem is inconsistent with the
no-signalling condition.
* The no-cloning and the no-deleting theorems point to the conservation of quantum information.
* A stronger version of the no-cloning theorem and the no-deleting theorem provide permanence to quantum information. To create a copy one must import the information from some part of the universe and to delete a state one needs to export it to another part of the universe where it will continue to exist.
See also
*
No-broadcast theorem
In physics, the no-broadcasting theorem is a result of quantum information theory. In the case of pure quantum states, it is a corollary of the no-cloning theorem. The no-cloning theorem for pure states says that it is impossible to create two ...
*
No-cloning theorem In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theore ...
*
No-communication theorem
In physics, the no-communication theorem or no-signaling principle is a no-go theorem from quantum information theory which states that, during measurement of an entangled quantum state, it is not possible for one observer, by making a measurem ...
*
No-hiding theorem
The no-hiding theorem states that if information is lost from a system via decoherence, then it moves to the subspace of the environment and it cannot remain in the correlation between the system and the environment. This is a fundamental consequen ...
Quantum no-hiding theorem experimentally confirmed for first time. Mar 07, 2011 by Lisa Zyga
/ref>
* Quantum cloning
Quantum cloning is a process that takes an arbitrary, unknown quantum state and makes an exact copy without altering the original state in any way. Quantum cloning is forbidden by the laws of quantum mechanics as shown by the no cloning theorem, wh ...
* Quantum entanglement
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
* Quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both th ...
* Quantum teleportation
* Uncertainty principle
In quantum mechanics, the uncertainty principle (also known as Heisenberg's uncertainty principle) is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physic ...
References
{{DEFAULTSORT:No-Deleting Theorem
Quantum information science
Theorems in quantum mechanics
No-go theorems